# Catalan's Conjecture

## Theorem

The only solution to the Diophantine equation:

$x^a - y^b = 1$

for $a, b > 1$ and $x, y > 0$, is:

$x = 3, a = 2, y = 2, b = 3$

## Also known as

This result is also known as Mihăilescu's Theorem, for Preda V. Mihăilescu.

## Source of Name

This entry was named for Eugène Charles Catalan.

## Historical Note

Catalan's Conjecture was first put forward by Eugène Charles Catalan in $1844$.

It was proven in $2002$ by Preda V. Mihăilescu.